cp's OEIS Frontend

A123234 Number of n X n Latin squares up to row and column permutation (or "RC-equivalence").

%I A123234 #15 Jul 21 2020 02:48:22
%S A123234 1,1,1,4,16,1868,2420400,66915816462
%N A123234 Number of n X n Latin squares up to row and column permutation (or "RC-equivalence").
%C A123234 _Brendan McKay_ writes: (Start)
%C A123234 "It would be possible to find the counts for n=9 and n=10 using the method of my paper in JCD [see link below]. For n=10 it is probably a 24-digit number. I'll explain the method I used. See the paper above for terminology.
%C A123234 "Is(L) is the autotopism group. Also define the group RC(L) of all autotopisms for which the symbols component is the identity. For any Latin square L we have:
%C A123234 "The isotopy class containing L contains (n!)^3/|Is(L)| squares.
%C A123234 "The RC-equivalence class containing L contains (n!)^2/|RC(L)| squares.
%C A123234 "If L and L' are isotopic then |RC(L)| = |RC(L')|. Therefore the number of RC-equivalence classes in the isotopy class of L is n!*|RC(L)|/|Is(L)|. I modified an existing program slightly to find |RC(L)|/|Is(L)|. and applied it to one square from each isotopy class. The sum of n!*|RC(L)|/|Is(L)| is the total number of RC-equivalence classes. " (End)
%D A123234 Dan R. Eilers, Phil A. Sallee, The number of Latin squares up to row and column permutation, Poster Session, Harvey Mudd College Mathematics Conference on Enumerative Combinatorics (2006) (for terms 1 to 7)
%D A123234 Brendan D. McKay, private communication (2006) (for term 8)
%H A123234 B. D. McKay, A. Meynert, W. Myrvold, (2007), <a href="http://users.cecs.anu.edu.au/~bdm/papers/ls_final.pdf">Small latin squares, quasigroups, and loops</a>, J. Combin. Designs, 15 (2007), 98-119. <a href="https://doi.org/10.1002/jcd.20105">doi:10.1002/jcd.20105</a>
%H A123234 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
%e A123234 01234 => 20413 => 01234
%e A123234 13042 => 01234 => 14320
%e A123234 24310 => 32041 => 20413
%e A123234 30421 => 43102 => 32041
%e A123234 42103 => 14320 => 43102
%e A123234 The first square is transformed by permuting columns; the 2nd square is transformed by permuting rows.
%e A123234 Both the first and 3rd square are in reduced form, so are considered equivalent by row/col permutation.
%Y A123234 Cf. A000315, A002724, A058163.
%K A123234 more,nice,nonn
%O A123234 1,4
%A A123234 _Dan Eilers_, Oct 06 2006